| L(s) = 1 | + 9·3-s − 69·5-s + 81·9-s + 123·11-s − 4·13-s − 621·15-s + 1.77e3·17-s − 1.39e3·19-s − 1.53e3·23-s + 1.63e3·25-s + 729·27-s − 3.61e3·29-s + 7.29e3·31-s + 1.10e3·33-s + 7.64e3·37-s − 36·39-s + 1.00e4·41-s − 9.75e3·43-s − 5.58e3·45-s − 1.76e4·47-s + 1.59e4·51-s + 4.19e3·53-s − 8.48e3·55-s − 1.25e4·57-s − 2.01e4·59-s + 3.46e4·61-s + 276·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.23·5-s + 1/3·9-s + 0.306·11-s − 0.00656·13-s − 0.712·15-s + 1.49·17-s − 0.887·19-s − 0.605·23-s + 0.523·25-s + 0.192·27-s − 0.798·29-s + 1.36·31-s + 0.176·33-s + 0.917·37-s − 0.00379·39-s + 0.932·41-s − 0.804·43-s − 0.411·45-s − 1.16·47-s + 0.860·51-s + 0.205·53-s − 0.378·55-s − 0.512·57-s − 0.752·59-s + 1.19·61-s + 0.00810·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 69 T + p^{5} T^{2} \) |
| 11 | \( 1 - 123 T + p^{5} T^{2} \) |
| 13 | \( 1 + 4 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1776 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1396 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1536 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3615 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7295 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7640 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10032 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9754 T + p^{5} T^{2} \) |
| 47 | \( 1 + 17622 T + p^{5} T^{2} \) |
| 53 | \( 1 - 4197 T + p^{5} T^{2} \) |
| 59 | \( 1 + 20133 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34646 T + p^{5} T^{2} \) |
| 67 | \( 1 + 41386 T + p^{5} T^{2} \) |
| 71 | \( 1 - 30762 T + p^{5} T^{2} \) |
| 73 | \( 1 + 80254 T + p^{5} T^{2} \) |
| 79 | \( 1 + 98593 T + p^{5} T^{2} \) |
| 83 | \( 1 - 73407 T + p^{5} T^{2} \) |
| 89 | \( 1 + 106554 T + p^{5} T^{2} \) |
| 97 | \( 1 + 161185 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.472780349722819105375507965343, −8.291378142792433580520357063371, −7.942694541832437524226161670575, −6.99740261878101501774604544835, −5.85991098454893146188578138957, −4.47729080290841038212396186320, −3.77449886527830749916033185491, −2.79586895737800800557599297989, −1.32341777343219013417771990053, 0,
1.32341777343219013417771990053, 2.79586895737800800557599297989, 3.77449886527830749916033185491, 4.47729080290841038212396186320, 5.85991098454893146188578138957, 6.99740261878101501774604544835, 7.942694541832437524226161670575, 8.291378142792433580520357063371, 9.472780349722819105375507965343
